Clubsuit
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In mathematics, and particularly in axiomatic set theory, ♣S (clubsuit) is a family of combinatorial principles that are weaker version of the corresponding ◊S; it was introduced in 1975 by A. Ostaszewski.
Definition
For a given cardinal number κ and a stationary set S ⊆ κ, ♣S is the statement that there is a sequence <math>\left\langle A_\delta: \delta \in S\right\rangle<math> such that
- every Aδ is a subset of δ
- for every unbounded subset A ⊆ κ, there is a δ so that Aδ ⊆ A
<math>\clubsuit_{\omega_1}<math> is usually written as just ♣.
♣ and ◊
It is clear that ◊ ⇒ ♣, and A. J. Ostaszewski showed in 1975 that ♣ + CH ⇒ ◊; however, Saharon Shelah gave proof in 1980 that there exists a model of ♣ in which CH does not hold, so ♣ and ◊ are not equivalent (since ◊ ⇒ CH).
References
- A. J. Ostaszewski, On countably compact perfectly normal spaces, Journal of London Mathematical Society, 1975 (2) 14, pp. 505-516.
- S. Shelah, Whitehead groups may not be free, even assuming CH, II, Israel Journal of Mathematics, 1980 (35) pp. 257-285.